Theory for Concentration and Solvency Effects in Size-Exclusion Chromatography of Polymers - Fleer, G. J; Skvortsov, A. M (Macromolecules, 2005)
A simple analytical equation for the distribution coefficient K in size-exclusion chromatography (SEC) as a function of molar mass, concentration, and solvent quality is presented. The theory is based upon a modified Casassa equation, using a recently proposed mean-field relation for the depletion thickness δ, which for better than ϑ conditions reads 1/δ2 = 1/δ0 2 + 1/ξ2. Here δ0 is the well-known (chain-length-dependent) depletion thickness at infinite dilution, and ξ is the (concentration- and solvency-dependent) correlation length in the solution. Numerical lattice calculations for mean-field chains in slitlike pores of width D as a function of concentration are in quantitative agreement with our analytical equation, both for good solvents and in a ϑ solvent. Comparison of our mean-field theory with Monte Carlo data for the concentration dependence of K for self-avoiding chains shows qualitatively the same trends; moreover, our model can also be adjusted to obtain nearly quantitative agreement. The modified Casassa equation works excellently in the wide-pore regime (where K = 1 − 2δ/D) and gives an upper bound for the narrow-pore regime. In fact, the simple form K = 1 − 2δ/D for 2δ/D < 1 and K = 0 for 2δ/D > 1 gives a first estimate of concentration effects even in the narrow-pore regime. A more detailed analysis of interacting depletion layers in narrow pores shows that a different length scale δi (the “interaction distance”) enters, which in semidilute solutions is somewhat higher than δ, leading to a smaller K than that obtained with the wide-pore length scale δ. Predictions for the effects of chain length, solvency, and chain stiffness on the basis of our analytical equation are in accordance with Monte Carlo simulations.